## Geometry & Topology

### Dominating surface group representations and deforming closed anti-de Sitter $3$–manifolds

Nicolas Tholozan

#### Abstract

Let $S$ be a closed oriented surface of negative Euler characteristic and $M$ a complete contractible Riemannian manifold. A Fuchsian representation $j : π1(S) → Isom+(ℍ2)$ strictly dominates a representation $ρ: π1(S) → Isom(M)$ if there exists a $(j,ρ)$–equivariant map from $ℍ2$ to $M$ that is $λ$–Lipschitz for some $λ < 1$. In a previous paper by Deroin and Tholozan, the authors construct a map $Ψρ$ from the Teichmüller space $T (S)$ of the surface $S$ to itself and prove that, when $M$ has sectional curvature at most $− 1$, the image of $Ψρ$ lies (almost always) in the domain $Dom(ρ)$ of Fuchsian representations strictly dominating $ρ$. Here we prove that $Ψρ: T (S) → Dom(ρ)$ is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations $(j,ρ)$ from $π1(S)$ to $Isom+(ℍ2)$ with $j$ Fuchsian strictly dominating $ρ$. In particular, we obtain that its connected components are classified by the Euler class of $ρ$. The link with anti-de Sitter geometry comes from a theorem of Kassel, stating that those pairs parametrize deformation spaces of anti-de Sitter structures on closed $3$–manifolds.

#### Article information

Source
Geom. Topol., Volume 21, Number 1 (2017), 193-214.

Dates
Revised: 29 January 2016
Accepted: 29 February 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859132

Digital Object Identifier
doi:10.2140/gt.2017.21.193

Mathematical Reviews number (MathSciNet)
MR3608712

Zentralblatt MATH identifier
1361.30067

#### Citation

Tholozan, Nicolas. Dominating surface group representations and deforming closed anti-de Sitter $3$–manifolds. Geom. Topol. 21 (2017), no. 1, 193--214. doi:10.2140/gt.2017.21.193. https://projecteuclid.org/euclid.gt/1510859132

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